Flying spaghetti monsters: Are Godel's "incompleteness theorems" actually "important?"
Do they carry any social significance? In the end, do they even make sense?
We'll admit to being doubters on the last of those points. Consider a part of Rebecca Goldstein's book which we'll explore in more detail at some later point.
Goldstein's book, designed for general readers, appeared in 2005. It bore this title: Incompleteness: The Proof and Paradox of Kurt Godel.
In our view, the general reader won't likely emerge from this book with the ability to discuss these supposedly transplendent theorems. For ourselves, we were surprised by the way Goldstein, a philosophy professor, leaned on the concept of "paradox" in her discussions, not excluding this rumintaion on a famous "abstract object:"
GOLDSTEIN (page 91): Russell's paradox concerns the set of all sets that are not members of themselves. Sets are abstract objects that contain members, and some sets can be members of themselves. For example, the set of all abstract objects is a member of itself, since it is an abstract object. Some sets (most) are not members of themselves. For example, the set of all mathematicians is not itself a mathematician—it's an abstract object—and so is not a member of itself. Now we form the concept of the set of all sets that aren't members of themselves and we ask of ourselves: is it a member of itself?...Now we form the concept of the set of all sets that aren't members of themselves? But why in the world would we do that?
The paragraph continues from there. The reference to "Russell" is a reference to Lord Russell, eventual husband of Lady Ottoline—that is to say, to Bertrand Russell—who came up with this world-class groaner back in 1901.
When we first encountered Goldstein's book, it surprised us to think that a capable philosophy professor would still be trafficking in this antique hocus-pocus about these "abstract objects"—about "abstract objects" which may or may not be "members of themselves."
We were even more surprised to see her marveling about this pseudo-paradox, which is even more simple-minded:
"This very sentence is false."Good God! The later Wittgenstein returned to England hoping to remove these flying spaghetti monsters from the pseudo-discourse in which he himself had trafficked as the early Wittgenstein. We were surprised to see a ranking professor still shoveling these snowstorms around.
We'll discuss these matters in the weeks ahead, possibly next week. For ourselves, if Godel's theorems turn on piddle like this, we'll float the shocking possibility that they may not make any real sense.
We know it's shocking to hear such claims about the genius theorems Goldstein gushes about. Then again, this "greatest logicians since Aristotle" seems to have been mentally ill his entire life; eventually died of self-starvation; and believed all sorts of crazy idea, perhaps including the crazy idea that numbers and circles live "a perfect, timeless existence" somewhere, apparently in an "abstract" realm we can access through something resembling ESP.
Do the theorems of this unfortunate man actually make any sense? For now, we'll vote with the doubters. Meanwhile, when Jordan Ellenberg discussed Goldstein's book for Slate, he offered these remarks, among others:
ELLENBERG (3/10/05): In his recent New York Times review of Incompleteness, Edward Rothstein wrote that it’s “difficult to overstate the impact of Gödel’s theorem.” But actually, it’s easy to overstate it: Goldstein does it when she likens the impact of Gödel’s incompleteness theorem to that of relativity and quantum mechanics and calls him “the most famous mathematician that you have most likely never heard of.” But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein’s and Heisenberg’s contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.You can read the rest of what Ellenberg wrote. For now, we're just saying!
When our greatest logicians devote their lives to the antics of spaghetti monsters, should we be surprised by the sheer stupidity which obtains all over the national discourse engineered by corporate journalists? We'll be focusing on that question next week. For today, let's visit an early part of Goldstein's book, where she starts to get something right.
When Holt summarized Goldstein's book, he profiled the strangeness of Godel. Again, we ask you to marvel at the highlighted part of this pile:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind...[T]he members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.Sadly, strangely, possibly dumbly, the greatest minds in Europe were puzzling hard over this:
What makes a proposition like "2 + 2 = 4" true?Seriously though, folks! From 1901 right up through Godel's arrival at college, that's what our allegedly greatest minds were struggling to figure out!
We mention this for a reason. Near the start of her book, Goldstein gives a weirdly decent explanation of this potent conundrum. She speaks about a different fact—the fact that 5 + 7 = 12—but, as you can probably see, the basic logic of all such statements is pretty much the same.
Citizens, we encountered this same traditional groaner as college freshmen ourselves! How can we know that 7 + 5 = 12? Professor Nozick raised this "problem" in the introductory course, Phil 3: Problems in philosophy.
How do we know that 7 + 5 = 12? One wag in the back of the class dared to ask himself this:
Who is this "problem in philosophy" a problem for?Or words to that effect! On the world's most exalted comedy stages, we've occasionally recalled one subsequent discussion. We did so just a few years ago, with comedy-loving Clarence Page as an opening act:
PHILOSOPHICALLY TORTURED TEACHING ASSISTANT: Students, how can we know that 7 + 5 equals 12?Did that exchange really take place? Memory sometimes plays tricks. But we're fairly sure that we remember the paper we finally wrote on this topic, and it resembled the explanation Goldstein supplies early in her book.
INNOCENT FRESHMAN: Miss Cummings told us? In second grade?
FRUSTRATED TEACHING ASSISTANT (tearing his hair as he stares out the window, seeming to contemplate the abyss): No, no, students, you're missing my point! How do we know that 7 + 5 equals 12?
[Pregnant pause]
PUZZLED FRESHMAN: Same answer?
What makes a proposition like “2 + 2 = 4” true? Using a slightly tougher example, Goldstein offers this:
GOLDSTEIN (page 17): The rigor and certainty of the mathematician is arrived at a priori, meaning that the mathematician neither resorts to any observations in arriving at his or her mathematical insights nor do these mathematical insights, in and of themselves, entail observations, so that nothing we experience can undermine the grounds we have for knowing them. No experience would count as grounds for revising, for example, that 5 + 7 = 12. Were we to add up 5 things and 7 things, and get 13 things, we would recount. Should we still, after repeated recountings, get 13 things we would assume that one of the 12 things had split or that we were seeing double or dreaming or even going mad. The truth that 5 + 7 = 12 is used to evaluate counting experiences, not the other way around.Goldstein is on the right track. That said, and stating the obvious, it makes more sense to explore the logic of this conundrum through the simplest possible example: 1 + 1 = 2.
How do we know that 1 + 1 = 2? Simple! Among other factors, we would be strongly disinclined to accept alleged counterexamples! Let's think in terms of marbles.
"Two" is simply the name we give to the number of marbles you'll typically have if you start with one marble, then receive one additional marble. If you counted your marbles at that point and found you had three marbles, we would assume that you hadn't noticed the addition of the third marble. Beyond that, we wouldn't accept such counterexamples as these:
Haystack Calhoun does the math:What would we say to such counterexamples? We would say they aren't what we mean! In each case, that simply isn't what we mean when we say 1 + 1 = 2!
A farmer has one haystack. He adds to it a second haystack. He sees that he still has one (larger) haystack. The farmer declares that, at least on the farm, 1 + 1 = 1.
Porky Pig adds to the wealth:
A farmer has one (male) pig. He adds one (female) pig. Months later, he finds that he has eight pigs. The farmer declares that, at least on the farm, 1 + 1 = 8.
The evaporation monologues:
A chemist has one beaker of a chemical. He adds a second beaker of a different chemical. The beaker's contents go "poof" and all the liquid disappears. When he added the second beaker, he ended up with no beakers. The chemist declares that, at least in the lab, 1 + 1 = 0.
How do we know that 1 + 1 = 2? We know it because we know what we mean when we make the familiar statement. All other addition facts follow from there. No flying spaghetti monsters, abstract or not, need apply!
Goldstein made a decent play on page 17. In our view, her book goes downhill from there, biographical writing excluded.
People, one plus one equals two! As our greatest thinkers argued this point, war came to Europe again.
Next week: The guardians file
Now for the rest of the story: After we freshmen took Phil 3, we all decided to abandon philosophy as a major. Nozick, who was only 26 at the time, went on to become a huge star. (He was always very nice to us pitiful freshmen.)
We switched back after sophomore year. Historical inevitability seemed to take over from there.
There seems to be a philosophical bar fight going on between Bertrand Russell and Wittgenstein, with Somerby taking sides with Wittgenstein against Russell and thus against those philosophers who continued to grapple with Russell's issues, including Godel.
ReplyDeleteSomerby says:
"Good God! The later Wittgenstein returned to England hoping to remove these flying spaghetti monsters from the pseudo-discourse in which he himself had trafficked as the early Wittgenstein. We were surprised to see a ranking professor still shoveling these snowstorms around."
But Russell says about Wittgenstein:
"I have not found in Wittgenstein's Philosophical Investigations anything that seemed to me interesting and I do not understand why a whole school finds important wisdom in its pages. Psychologically this is surprising. The earlier Wittgenstein, whom I knew intimately, was a man addicted to passionately intense thinking, profoundly aware of difficult problems of which I, like him, felt the importance, and possessed (or at least so I thought) of true philosophical genius. The later Wittgenstein, on the contrary, seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary. I do not for one moment believe that the doctrine which has these lazy consequences is true. I realize, however, that I have an overpoweringly strong bias against it, for, if it is true, philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea-table amusement."
I believe that same laziness infected Somerby, who seems to reject Godel on Wittgenstein's say-so.
If this dispute is discussed by Goldstein, Somerby doesn't mention it.
Somerby once again claims that Godel's theorem has no practical importance. Philosophers and computer scientists disagree. He again implies that if there is a war going on, all non-practical endeavors must cease (did painters stop painting, authors stop writing?). He almost implies that the focus of philosophers on Godel's theorem permitted the war to happen because professors were distracted, looking elsewhere. He has said such things more explicitly -- he seems to think that professors have a duty to stop war and poverty and all social problems, but are frittering away their time on useless pursuits. Project much?
I find myself wondering why Somerby doesn't devote his posts to explaining Wittgenstein instead of attacking innocuous popular books on other thinkers. But maybe he is afraid his readers would skip straight to the chase and just go chasing Chomsky.
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Oh my god, Bob. I've loved your columns for over a decade, but you're so far ahead of your skis here that I'm surprised you can still stand up. Stop talking about Godel. The simple truth is that this material is way over your head. I'm not saying that you lack the intelligence, but you definitely lack the background.
ReplyDeleteSimply put, Godel is that godfather of computational theory. We would not be communicating in this way if it were not for him.
When you say stuff like this:
"Now we form the concept of the set of all sets that aren't members of themselves? But why in the world would we do that?"
...it becomes clear that you have no background in mathematics. I could answer that question for you, but it would take two semesters.
Listen, you can't learn this stuff from newspaper articles or books for a layman. You just can't! You're not going to get it intuitively. It all seems like pointless intellectual exercises to you because the relationship between this material and the life you lead is very indirect. It's real and important, though. Just because you can't understand something doesn't mean it's not significant.
I am begging you - stop this. You have no idea how foolish this looks to someone who understands the material. You have to learn to accept that some things are incomprehensible without the hard work it takes to understand it. You seem to balk hard at that idea, but the truth is that you're not even qualified to debate it. This is one of those cases where you don't know what you don't know.
It's really depressing for me to read this nonsense, because so often you have made insightful and intelligent statements about politics and education, and I've taken much of it to heart. But this is just the arrogance of a man who thinks he can comment on something highly complex without bothering to learn the basics.
Here's a test: the full flowering of the Incompleteness Theorem is expressed in the concept of NP-completeness. If you can read and understand this wikipedia page, and why it is central to understanding computers, then I was wrong and you're qualified to discuss this topic:
https://en.wikipedia.org/wiki/NP-completeness
Please, please, stop this nonsense.
This comment has been removed by the author.
DeleteUmm...yes. In a round-about way this reminds me of a great Errol Morris' documentary:Mr. Death: The Rise and Fall of Fred A. Leuchter, Jr. Where someone who has core competency in one subject decides that he can figure out anything just by bootstrapping himself from that core competency. It doesn't work this way. Bob doesn't seem to get it.
DeleteEverything seems to be based on the assumption that "since I can't intuit it, it must be bunk".
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DeleteLeroy, you deleted your comment, but I got an email for it and it was reasonable, so I shall respond. You asked if I got that Bob was criticizing the media. The answer is, yes and no, but I'm pushing back against both, here.
DeleteYes, I understand the Bob is, on one hand, criticizing the media's ability to explain Godel. But part of the point I keep making is that you can't expect it to do a good job at this. The material is just too dense and technical. So I feel he's really beating this to death, because there's really no chance for improvement, unless he wants to see the media to just stop trying.
But on the other hand, Bob isn't just questioning the reporting. Here's how he starts the post:
"Are Godel's "incompleteness theorems" actually "important?"
Do they carry any social significance? In the end, do they even make sense?
We'll admit to being doubters on the last of those points."
Perhaps noticing this was what caused you to delete your post. In any case, you shouldn't have felt so abashed, because at least you were very polite about your criticism. Maybe a tiny bit condescending, but I've faced far far worse online, so I don't hold that against you.
Ilya, you are very correct. This is a bit amusing to me because usually it's members of my profession - software engineers - who are accused of this. In fact, this is sometimes called "engineer's disease." I guess Bob is exhibiting reverse engineer's disease.
DeleteTo extend Somerby's scenario: The freshman sits in his philosophy class. He may or may not have done the assigned reading before coming to class. The professor explains key concepts and the freshman struggles mightily but just does not understand what is being said. What does he do at that point?
ReplyDelete1. The freshman blames the professor. To preserve his self-esteem, he changes his major, then comes back to it later after having found that there are one or two philosophers who he can understand without major effort. This is Somerby's path.
2. The freshman visits the teaching assistant during office hours and seeks more explanation. If that is unhelpful, he asks for additional reading. He backtracks by looking up precursor terms, definitions, ideas, previously assigned readings. He talks about the topic with others in the class to see how they understand it. He sticks with it until it begins to make some sense.
3. The freshman continues the course without understanding that material, in the hope that what comes after will be more comprehensible. Sometimes it is, often it isn't. He gets the highest grade he can on any exams (often C's at best) and floats through each successive class that way until graduation, whereupon he proclaims that he was a philosophy major. Many students do this.
4. He rebels against philosophy, declaring it a dead topic with a foolish focus that needs major revision. If he is glib, he invents an alternative school of thought and introduces it as the new philosophy. Bright but undisciplined students take this route. Serious philosophers are not distracted by it, but students will follow.
Somerby's problems with philosophy are not unique to that discipline. There are people who regularly declare statistics unnecessary to the social sciences (a revolt against quantitative approaches in sociology, for example), because math is hard. Disdain for what you cannot do is widespread.
"Disdain for what you cannot do is widespread."
ReplyDeleteReally well-put. This phrase reminds me that Bob is indulging in the sort of anti-intellectualism that he usually decries.
Bob definitely has a problem with philosophy, but he doesn't seem to realize that Godel was much more of a mathematician than philosopher. It's quite disenheartening to see him so out of his element.
I have read Bob for over a decade as well. I wish he would return to his regular blogging and not talk about philosophy. I think he may have some kind of point to make, but, I don't think this is the way to make it.
DeleteActually, if you look back over his blog, there is an anti-intellectual strain running through it for at least the past 10 years. It took the form of attacks on "professors". It never seemed to be just specific objections to specific professors to me at least.
DeleteThat's true about the anti-intellectualism. It was a bit more subtle, though, and to tell you the truth, I sometimes agreed with Bob. But this is a case where I actually know what he's talking about, and it's clear that he's 100% out to sea. So now I'm rethinking a lot of prior agreements I had with him. I'm not exaggerating when I say that I'm saddened by this.
DeleteThere are places I remember all my life, though some have changed, some forever not for better, some have gone and some remain.
ReplyDeletehttps://www.youtube.com/watch?v=YBcdt6DsLQA
Music is a picture, when it's recorded.
Deletehttps://www.youtube.com/watch?v=auLBLk4ibAk
Leroy
Forgot this one. Pay attention.
Deletehttps://www.youtube.com/watch?v=7xxgRUyzgs0
Leroy
IMHO the value of Russell's paradox is the following
ReplyDelete-- Lots of mathematics has practical value
-- Set theory is a handy tool when developing useful mathematics
-- Russell's paradox shows that one must take care in designing one's set theory structure and assumptions, or that structure will be flawed.
David, you are not a member of yourself, but your wienie is.
Delete[Bloomberg] "A survey commissioned by the Republican National Committee has led the party to a glum conclusion regarding President Donald Trump’s signature legislative achievement: Voters overwhelmingly believe his tax overhaul helps the wealthy instead of average Americans.
DeleteBy a 2-to-1 margin — 61 percent to 30 percent — respondents said the law benefits “large corporations and rich Americans” over “middle class families,” according to the survey, which was completed on Sept. 2 by the GOP firm Public Opinion Strategies and obtained by Bloomberg News.
The result was fueled by self-identified independent voters who said by a 36-point margin that large corporations and rich Americans benefit more from the tax law — a result that was even more lopsided among Democrats. Republican voters said by a 38-point margin that the middle class benefits more. …
… The RNC study says Americans worry the tax law will lead to cuts in Social Security and Medicare, concluding that “most voters believe that the GOP wants to cut back on these programs in order to provide tax breaks for corporations and the wealthy.” It attributes that finding to “a fairly disciplined Democrat attack against the recent tax cuts.”
Give it a rest, Bob! At least read Suzanne Langer's lucid and disarming textbook "An Introduction to Symbolic Logic" (https://www.amazon.com/Introduction-Symbolic-Logic-3rd/dp/0486601641) before you go on outgassing in the common room like some bright college freshman. At your age don't you realize how many "bright college freshmen" there are? Anyway, at least read Langer's book. She was a logician and philosopher who wrote well.
ReplyDeleteI find the question of "How do we know something is true?" to be interesting and fairly fundamental. The fact that it starts with something basic like 1+1 seems logical to me since more complex "truths" are built upon basic ones.
ReplyDeleteThe sarcastic freshman answer was just the Bible believers mantra restated.
1. the teacher said it
2. I believe it
3. that settles it
Although, as I mentioned before, that was mostly the way I learned - you memorize the terms, you memorize the dates or other key "facts" and you memorize the formulas. You get good grades.
In some ways though, I seem to have been the dumbest kid in school, because I took it seriously. I was gullible enough to believe the lie that "school is important, good grades are important".
My classmates, and my siblings had more fun and also did "better" in life that I with all my studying.
One meta question that most eighth grade algebra students ask is "Why do we need to know this?" or "Why are we studying this?" I gave up on the idea of teaching algebra because I agreed with the students - most of you really don't.
Even less so then do we need to think about why we know that 1+1=2
However, to say that we know it is true because we understand what we mean by it, does not seem like a valid answer. Consider other simple statements.
The capital of South Dakota is Bismarck.
The capital of South Dakota is Huron.
Pierre is the capital of Montana.
We know what we mean by all of those statements, but none of them are true. Simply understanding the meaning of a statement does not make it true.
I would say that we do, in theory, verify our math by experience. 5+7=12 because we can observe that. It is observable and repeatable, and experience could, in theory prove it wrong. If today, after 56 years of experience where 5+7=12 I added 5 of something to 7 of something and came up with 13, and repeated it again and again and again. And if everybody in the world got the same result, then we would have to change our opinion to fit the facts. That reality had somehow changed in some way.
Such a change would not fit my experience of the last 56 years or my dad's experience of the last 84 years, but it would be the new reality.
At the lower levels of schooling though, I would say that much of what we "know" is really based on that faith. The teacher (or a book) said it, so we believe it, or not, and for many of us we do not care one way or the other. If we wanted to know what the capital of South Dakota is, we would consult an atlas or an encyclopedia - basically an authority which we have faith in.
But do we care? Should we care? My alma mater is having another induction into its "Hall of Fame" this week. However, it is NOT a hall of fame for any kind of academic accomplishment. They are not inducting somebody who scored a 36 on the ACT in 1975 and had a 30 year career teaching physics at the school of mines.
Nope, there is no such hall of fame for such students, if they did exist. The hall of fame is for sports. So clearly, in our society, this equation is true - algebra < golf.
I am fascinated by epistomology, the investigation of what distinguishes justified belief from opinion. I looked at my work as an actuary as a kind of applied epistomology. When I recommended a certain premium rate based on a certain projected loss cost, how did one know that my projected loss cost was correct or appropriate? Proper statistical analyses would lead to correct projections more often than improper analyses would.
DeleteA survey commissioned by the Republican National Committee has led the party to a glum conclusion regarding President Donald Trump’s signature legislative achievement: Voters overwhelmingly believe his tax overhaul helps the wealthy instead of average Americans.
DeleteBy a 2-to-1 margin — 61 percent to 30 percent — respondents said the law benefits “large corporations and rich Americans” over “middle class families,” according to the survey, which was completed on Sept. 2 by the GOP firm Public Opinion Strategies and obtained by Bloomberg News.
The result was fueled by self-identified independent voters who said by a 36-point margin that large corporations and rich Americans benefit more from the tax law — a result that was even more lopsided among Democrats. Republican voters said by a 38-point margin that the middle class benefits more. …
… The RNC study says Americans worry the tax law will lead to cuts in Social Security and Medicare, concluding that “most voters believe that the GOP wants to cut back on these programs in order to provide tax breaks for corporations and the wealthy.” It attributes that finding to “a fairly disciplined Democrat attack against the recent tax cuts.”
In computer science there's something called "halt theorem", which is closely related to Goedel's incompleteness theorem. Halt theorem has some very important implications in practical computing. Bob, please confine yourself to the media criticism, at which you excel.
ReplyDeleteYou don't have to learn anything if you are OK with living a limited life. If you want more, you need to do more. You get what you put into life. If all you want to do is have fun and enjoy yourself, don't bother with anything except drugs. Don't worry about making a living, you can pay for your drugs with sex.
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ReplyDeleteI want to tell all the women/men out there who have a similar situation like mine,that the world is not over YET they should dry up their tears and contact this great man and their problem will be gone or are you also having other problems you can also contact Dr Alexzander, here is how you can contact him alexzanderhightemple@gmail.com or contact him via his whatsapp phone number +2348075823891.
Thank Dr Alexzander for everything you did in my marriage.
Thanks
Caroline Grilli
Upyernoz.blogspot.com makes a good point:
ReplyDelete"The main reason that people think that Brett Kavanaugh is qualified to sit on the Supreme Court is because proponents of the nomination keep saying that he is well-qualified. But nothing in his actual record makes him a legal superstar. Instead, whether we are talking about his time as a student, or working on the now-disgraced Starr investigation, or working for the failed Bush administration, or his lackluster recent years on the Court of Appeals, I am just not seeing a particularly talented individual, or any remarkable intellect. Instead, Kavanaugh's story seems to be the story of how in America privilege delivers. No matter what you do, if you are born into a privileged class, you will fall up throughout your career.
I submit this makes Kavanaugh particularly unqualified for the high court. He had many more opportunities to do great things that most and yet he didn't do anything great or even mildly remarkable.1 There is nothing in Kavanaugh's history that suggest he should be on the most important court in the country. And that is without considering the the other reasons to oppose his nomination (i.e. the perjury, the attempted rape, the gambling debts that were paid under mysterious circumstances, and the massive amounts of withheld records). Even if you push all that aside, Kavanaugh is a terrible nominee, eerie parallels to the history of privilege of our own "falling up" president."
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Thank you Thank You Very much Doctor Otonokpo for making my ex boyfriend come back to me. I am Cordelia Sandra from Brazil and i am putting this testimony here too because i want to share my testimony of how i was helped by Doctor Otonokpo within 48 hours of contacting him. Yes, it was last week my ex boyfriend returned to me after i contacted Doctor Otonokpo. My boyfriend was always going back to meet his ex girlfriend because he never really left her. Her name was Sophie. I didn't know how it happened one day after breakfast that i saw him looking at his ex girlfriend's picture on Facebook and I flared at him that he doesn't care about him and he was with me and still thinking about his ex although we have been dating for 6 months. He stormed at me and left the house and never returned. I was heartbroken and wanted him to come back. I was in a nightclub with friend one evening that I saw him with Sophie there, I was humiliated that night and I regretted going there only to see him there. I went online after some days and found Doctor Otonokpo and read about him and I contacted him to help me get him back. I must say that within 48 hours, my boyfriend came back to me and pleaded for leaving me. Is this how spell works so fast? Please, if you want help, contact Doctor Otonokpo too to help you at otonokpotemple@gmail.com
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